In this paper, we focus on recursive estimation of UBSR, in a setting where data arrives in an online fashion. Estimation of UBSR has immediate applications in financial portfolio optimization, cf. [1]}. Stochastic approximation [2]}, [3]} is a procedure that is well-suited for the purpose of online estimation. In the context of UBSR estimation, our main contribution is the non-asymptotic analysis of a stochastic approximation-based estimation scheme. We cast the estimation of UBSR as a stochastic root finding problem, and derive `finite-sample' bounds for this scheme. Our analysis assumes that the underlying objective satisfies a monotonicity condition. If the monotonicity parameter is known and is used in setting the step-size, the algorithm results in an \(O(1/n)\)  rate of error decay. We also develop another variant that employs a universal step-size, and results in a \(O(1/n^\alpha )\)  rate, where \(0<\alpha <1\) . These non-asymptotic results are obtained under similar technical assumptions as in [4]}, [1]} — specifically, a finite second moment condition on the loss distribution. If the loss distribution is sub-Gaussian, we also obtain a `high probability' result for the concentration of the approximation error.
